Loeb Measures in Practice: Recent Advances

1. Loeb Measures1.1 Introduction1.2 Nonstandard Analysis1.2.1 The hyperreals1.2.2 The nonstandard universe1.2.3 N1-saturation1.2.4 Nonstandard topology1.3 Construction of Loeb Measures1.3.1 Example: Lebesgue measure1.3.2 Example: Haar measure1.3.3 Example: Wiener measure1.3.4 Loeb measurable functions 1.4 Loeb Integration Theory1.5 Elementary Applications1.5.1 Lebesgue integration1.5.2 Peano's Existence Theorem1.5.3 Ito integration and stochastic differential equations 2 Stochastic Fluid Mechanics2.1 Introduction2.1.1 Function spaces2.1.2 Functional formulation of the Navier-Stokes equations2.1.3 Definition of solutions to the stochastic Navier-Stokes equations2.1.4 Nonstandard topology in Hilbert spaces2.2 Solution of the Deterministic Navier-Stokes Equations2.2.1 Uniqueness2.3 Solution of the Stochastic Navier-Stokes Equations2.3.1 Stochastic Flow2.3.2 Nonhomogeneous stochastic Navier-Stokes equations2.4 Stochastic Euler Equations2.5 Statistical Solutions 2.5.1 The Foias equation2.5.2 Construction of statistical solutions using Loeb measures2.5.3 Measures by nonstandard densities2.5.4 Construction of statistical solutions using nonstandard densities2.5.5 Statistical solutions for stochastic Navier-Stokes equations2.6 Attractors for the Navier-Stokes Equations2.6.1 Introduction2.6.2 Nonstandard attractors and standard attractors2.6.3 Attractors for 3-dimensional Navier-Stokes equations2.7 Measure Attractors for Stochastic Navier-Stokes Equations2.8 Stochastic Attractors for Navier-Stokes Equations2.8.1 Stochastic attractors2.8.2 Existence of a stochastic attractor for the Navier-Stokes equations2.9 Attractors for the 3-dimensional Stochastic Navier-Stokes Equations3. Stochastic Calculus of Variations3.1 Introduction3.1.1 Notation3.2 Flat Integral Representation of Wiener Measure3.3 The Wiener Sphere3.4 Brownian Motion on the Wiener Sphere and the Infinite Dimensional Ornstein-Uhlenbeck Process3.5 Malliavin Calculus3.5.1 Notation and preliminaries3.5.2 The